On computable numbers, with an application to the Entscheidungsproblem; On computable numbers, with an application to the Entscheidungsproblem. A correction.

London: C.F. Hodgson and Son, 1936-37.

First edition, journal issue in the original printed wrappers (main paper). “‘On Computable Numbers’ is regarded as the founding publication of the modern science of computing. It contributed vital ideas to the development, in the 1940s, of the electronic stored-programme digital computer. ‘On Computable Numbers’ is the birthplace of the fundamental principle of the modern computer, the idea of controlling the machine's operations by means of a programme of coded instructions stored in the computer's memory. In addition Turing charted areas of mathematics lying beyond the scope of the Turing machine. He proved that not all precisely stated mathematical problems can be solved by computing machines. One such is the Entscheidungsproblem or ‘decision problem’ … In this one article, Turing ushered in both the modern computer and the mathematical study of the uncomputable” (Copeland, The Essential Turing, p. 6). The outstanding problem of mathematical logic at the time, the Entscheidungsproblem, posed by David Hilbert in 1928, asks whether there is an algorithm that can determine whether any given mathematical statement is true or not. “In the long view of intellectual history, I believe universal computation will stand as the single most important idea to emerge in the twentieth century. And this paper is where it first appeared with clarity” (Stephen Wolfram, in Alan Turing. His Work and Impact, p. 44). “Inspired by the human computer (i.e., the human engaged in computation), Turing described a notional machine that could read and write symbols along a segmented tape. The machine itself would be capable of assuming various internal states that, together with the input of a single symbol along the tape, could lead to a few primitive atomic actions. Based on the state and the current symbol, each configuration specifies a change (or not) of symbol, a move right or left, and a next state. Working under some straightforward assumptions about the finite and discrete nature of the machine, Turing was able to demonstrate the wide range of numbers (equivalently, the wide class of functions) that could be computed and, moreover, able to specify a single machine, the universal machine, that would be capable of simulating the computations of any such machine. Turing's characterization has come to be seen as a more compelling account of what it means to be effective, mechanical, or algorithmic than any of the various extensionally equivalent formulations offered by his contemporaries” (New Dictionary of Scientific Biography, vol. 7, p. 83). The ‘Correction’ was published in order to remove some formal errors made in the first paper pointed out by the Swiss mathematician Paul Bernays. ABPC/RBH list only two copies with all three parts in the original printed wrappers, the most recent being the Richard Green copy (Christie’s, June 17, 2013, $182,500).

“In the spring of 1935 – at the time of von Neumann's visit to Cambridge – Turing was attending Max Newman’s lectures on the foundations of mathematics when the Entscheidungsproblem first attracted his attention. Hilbert’s challenge aroused Turing’s instinct that mathematical questions resistant to strictly mechanical procedures could be proved to exist.

“Turing’s argument was straightforward – as long as you threw out all assumptions and started fresh. “One of the facets of extreme originality is not to regard as obvious the things that lesser minds call obvious,” says I. J. (Jack) Good, who served as an assistant to Turing (then referred to as “Prof”) during World War II. Originality can be more important than intelligence, and according to Good, Turing constituted proof. “Henri Poincare did quite badly at an intelligence test, and Prof also was only about halfway up the undergraduate scale when he took such a test.” Had Turing more closely followed the work of Alonzo Church or Emil Post, who anticipated his results, his interest might have taken a less original form. “The way in which he uses concrete objects such as exercise books and printer’s ink to illustrate and control the argument is typical of his insight and originality,” says colleague Robin Gandy. “Let us praise the uncluttered mind.”

“A function is computable, over the domain of the natural numbers (0, 1, 2, 3, …), if there exists a finite sequence of instructions (or algorithm) that prescribes exactly how to list the value of the function at f(0) and, for any natural number n, at f(n + 1). Turing approached the question of computable functions in the opposite direction, from the point of view of the numbers produced as a result. “According to my definition,” he explained, “a number is computable if its decimal can be written down by a machine.”

“Turing began with the informal idea of a computer – which in 1935 meant not a calculating machine but a human being, equipped with pencil, paper, and time. He then substituted unambiguous components until nothing but a formal definition of “computable” remained. Turing’s machine (which he termed an LCM, or Logical Computing Machine) thus consisted of a black box (as simple as a typewriter or as complicated as a human being) able to read and write a finite alphabet of symbols to and from a finite but unbounded length of paper tape – and capable of changing its own “m-configuration,” or “state of mind.”

“We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions … which will be called ‘m-configurations’,” Turing wrote. “The machine is supplied with a ‘tape’ (the analogue of paper) running through it, and divided into sections (called ‘squares’) each capable of bearing a ‘symbol.’ At any moment there is just one square … which is ‘in the machine’ … However, by altering its m-configuration the machine can effectively remember some of the symbols which it has ‘seen.’ … In some of the configurations in which the scanned square is blank (i.e., bears no symbol) the machine writes down a new symbol on the scanned square; in other configurations it erases the scanned symbol. The machine may also change the square which is being scanned, but only by shifting it one place to right or left. In addition to any of these operations the m-configuration may be changed.”

“Turing introduced two fundamental assumptions: discreteness of time and discreteness of state of mind. To a Turing machine, time exists not as a continuum, but as a sequence of changes of state. Turing assumed a finite number of possible states at any given time. “If we admitted an infinity of states of mind, some of them will be ‘arbitrarily close’ and will be confused,” he explained. “The restriction is not one which seriously affects computation, since the use of more complicated states of mind can be avoided by writing more symbols on the tape.”

“The Turing machine thus embodies the relationship between an array of symbols in space and a sequence of events in time. All traces of intelligence were removed. The machine can do nothing more intelligent at any given moment than make a mark, erase a mark, and move the tape one square to the right or to the left. The tape is not infinite, but if more tape is needed, the supply can be counted on never to run out. Each step in the relationship between tape and Turing machine is determined by an instruction table listing all possible internal states, all possible external symbols, and, for every possible combination, what to do (write or erase a symbol, move right or left, change the internal state) in the event that combination comes up. The Turing machine follows instructions and never makes mistakes. Complicated behavior does not require complicated states of mind. By taking copious notes, the Turing machine can function with as few as two internal states. Behavioral complexity is equivalent whether embodied in complex states of mind (m-configurations) or complex symbols (or strings of simple symbols) encoded on the tape.

“It took Turing only eleven pages of ‘On Computable Numbers’ to arrive at what became known as Turing’s Universal Machine. “It is possible to invent a single machine which can be used to compute any computable sequence,” he announced. The Universal Machine, when provided with a suitably encoded description of some other machine, executes this description to produce equivalent results. All Turing machines, and therefore all computable functions, can be encoded by strings of finite length. Since the number of possible machines is countable but the number of possible functions is not, noncomputable functions (and what Turing referred to as “uncomputable numbers”) must exist.

“Turing was able to construct, by a method similar to Gödel’s, functions that could be given a finite description but could not be computed by finite means. One of these was the halting function: given the number of a Turing machine and the number of an input tape, it returns either the value 0 or the value 1 depending on whether the computation will ever come to a halt. Turing called the configurations that halt ‘circular’ and the configurations that keep going indefinitely ‘circle free,’ and demonstrated that the unsolvability of the halting problem implies the unsolvability of a broad class of similar problems, including the Entscheidungsproblem. Contrary to Hilbert’s expectations, no mechanical procedure can be counted on to determine the provability of any given mathematical statement in a finite number of steps. This put a halt to the Hilbert program, while Hitler's purge of German universities put a halt to Göttingen’s position as the mathematical center of the world, leaving a vacuum for Turing’s Cambridge, and von Neumann’s Princeton, to fill.

“After a full year of work, Turing gave Newman a draft of his paper in April of 1936. “Max’s first sight of Alan’s masterpiece must have been a breathtaking experience, and from this day forth Alan became one of Max's principle protégés,” says William Newman, Max's son. Max Newman lobbied for the publication of ‘On Computable Numbers, with an Application to the Entscheidungsproblem,’ in the Proceedings of the London Mathematical Society, and arranged for Turing to go to Princeton to work with Alonzo Church. “This makes it all the more important that he should come into contact as soon as possible with the leading workers on this line, so that he should not develop into a confirmed solitary,” Newman wrote to Church.

“Turing arrived in Princeton carrying his sextant, and stretching his resources to survive on his King’s College fellowship (of £300) for the year. The page proofs of ‘On Computable Numbers’ arrived by mail from London on October 3. “It should not be long now before the paper comes out,” he wrote to his mother on October 6. The publication of ‘On Computable Numbers’ (on November 30, 1936) went largely unnoticed. “I was disappointed by its reception here,’ Turing wrote to his mother in February 1937, adding that “I don’t much care about the idea of spending a long summer in this country.” Only two requests for reprints came in. Engineers avoided Turing’s paper because it appeared entirely theoretical, and theoreticians avoided it because of the references to paper tape and machines …

“In March of I937, Alonzo Church reviewed ‘On Computable Numbers’ in the Journal of Symbolic Logic, and coined the term Turing machine. “Computability by a Turing machine,” wrote Church, “has the advantage of making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately.” Church’s thesis – equating computability with effective calculability – would be the Church-Turing thesis from then on.

“Even Gödel, who dismissed most attempts to strengthen his own results, recognized the Church-Turing thesis as a major advance. “With this concept one has for the first time succeeded in giving an absolute definition … not depending on the formalism chosen,” he admitted in 1946. Before Church and Turing, the definition of mechanical procedure was limited by the language in which the concept was defined. “For the concept of computability however … the situation is different,” Gödel observed. “By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion.”

“It is difficult today to realize how bold an innovation it was to introduce talk about paper tapes and patterns punched in them, into discussions of the foundations of mathematics,” Max Newman recalled in 1955. For Turing, the next challenge was to introduce mathematical logic into the foundations of machines. “Turing’s strong interest in all kinds of practical experiment made him even then interested in the possibility of actually constructing a machine on these lines.”

“The title ‘On Computable Numbers’ (rather than ‘On Computable Functions’) signaled a fundamental shift. Before Turing, things were done to numbers. After Turing, numbers began doing things. By showing that a machine could be encoded as a number, and a number decoded as a machine, ‘On Computable Numbers’ led to numbers (now called ‘software’) that were ‘computable’ in a way that was entirely new” (Dyson, Turing’s Cathedral, pp. 246-250).

Origins of Cyberspace 394; Tomash and Williams T61, T62.

Pp. 230-265 in Proceedings of the London Mathematical Society, series 2, vol. 42, part 3, November 30, 1936 & part 4, December 23, 1936; [with:] ‘... A correction,’ pp. 544-546 in ibid., vol. 43. Three vols., large 8vo, pp. 161-240; 241-320; [iv], 546. The two journal issues in original printed wrappers, preserved in a cloth folding case (remnants of publisher’s printed label on front wrapper of part 3, edges of wrappers of part 4 a little dust-soiled); the journal volume in (probably publisher’s) green cloth, covers ruled in blind, spine lettered in gilt, virtually mint.

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On computable numbers, with an application to the Entscheidungsproblem; On computable numbers, with an application to the Entscheidungsproblem. A correction.

London: C.F. Hodgson and Son, 1936-37.

First edition, journal issue in the original printed wrappers (main paper). “‘On Computable Numbers’ is regarded as the founding publication of the modern science of computing. It contributed vital ideas to the development, in the 1940s, of the electronic stored-programme digital computer. ‘On Computable Numbers’ is the birthplace of the fundamental principle of the modern computer, the idea of controlling the machine's operations by means of a programme of coded instructions stored in the computer's memory. In addition Turing charted areas of mathematics lying beyond the scope of the Turing machine. He proved that not all precisely stated mathematical problems can be solved by computing machines. One such is the Entscheidungsproblem or ‘decision problem’ … In this one article, Turing ushered in both the modern computer and the mathematical study of the uncomputable” (Copeland, The Essential Turing, p. 6). The outstanding problem of mathematical logic at the time, the Entscheidungsproblem, posed by David Hilbert in 1928, asks whether there is an algorithm that can determine whether any given mathematical statement is true or not. “In the long view of intellectual history, I believe universal computation will stand as the single most important idea to emerge in the twentieth century. And this paper is where it first appeared with clarity” (Stephen Wolfram, in Alan Turing. His Work and Impact, p. 44). “Inspired by the human computer (i.e., the human engaged in computation), Turing described a notional machine that could read and write symbols along a segmented tape. The machine itself would be capable of assuming various internal states that, together with the input of a single symbol along the tape, could lead to a few primitive atomic actions. Based on the state and the current symbol, each configuration specifies a change (or not) of symbol, a move right or left, and a next state. Working under some straightforward assumptions about the finite and discrete nature of the machine, Turing was able to demonstrate the wide range of numbers (equivalently, the wide class of functions) that could be computed and, moreover, able to specify a single machine, the universal machine, that would be capable of simulating the computations of any such machine. Turing's characterization has come to be seen as a more compelling account of what it means to be effective, mechanical, or algorithmic than any of the various extensionally equivalent formulations offered by his contemporaries” (New Dictionary of Scientific Biography, vol. 7, p. 83). The ‘Correction’ was published in order to remove some formal errors made in the first paper pointed out by the Swiss mathematician Paul Bernays. ABPC/RBH list only two copies with all three parts in the original printed wrappers, the most recent being the Richard Green copy (Christie’s, June 17, 2013, $182,500).

“In the spring of 1935 – at the time of von Neumann's visit to Cambridge – Turing was attending Max Newman’s lectures on the foundations of mathematics when the Entscheidungsproblem first attracted his attention. Hilbert’s challenge aroused Turing’s instinct that mathematical questions resistant to strictly mechanical procedures could be proved to exist.

“Turing’s argument was straightforward – as long as you threw out all assumptions and started fresh. “One of the facets of extreme originality is not to regard as obvious the things that lesser minds call obvious,” says I. J. (Jack) Good, who served as an assistant to Turing (then referred to as “Prof”) during World War II. Originality can be more important than intelligence, and according to Good, Turing constituted proof. “Henri Poincare did quite badly at an intelligence test, and Prof also was only about halfway up the undergraduate scale when he took such a test.” Had Turing more closely followed the work of Alonzo Church or Emil Post, who anticipated his results, his interest might have taken a less original form. “The way in which he uses concrete objects such as exercise books and printer’s ink to illustrate and control the argument is typical of his insight and originality,” says colleague Robin Gandy. “Let us praise the uncluttered mind.”

“A function is computable, over the domain of the natural numbers (0, 1, 2, 3, …), if there exists a finite sequence of instructions (or algorithm) that prescribes exactly how to list the value of the function at f(0) and, for any natural number n, at f(n + 1). Turing approached the question of computable functions in the opposite direction, from the point of view of the numbers produced as a result. “According to my definition,” he explained, “a number is computable if its decimal can be written down by a machine.”

“Turing began with the informal idea of a computer – which in 1935 meant not a calculating machine but a human being, equipped with pencil, paper, and time. He then substituted unambiguous components until nothing but a formal definition of “computable” remained. Turing’s machine (which he termed an LCM, or Logical Computing Machine) thus consisted of a black box (as simple as a typewriter or as complicated as a human being) able to read and write a finite alphabet of symbols to and from a finite but unbounded length of paper tape – and capable of changing its own “m-configuration,” or “state of mind.”

“We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions … which will be called ‘m-configurations’,” Turing wrote. “The machine is supplied with a ‘tape’ (the analogue of paper) running through it, and divided into sections (called ‘squares’) each capable of bearing a ‘symbol.’ At any moment there is just one square … which is ‘in the machine’ … However, by altering its m-configuration the machine can effectively remember some of the symbols which it has ‘seen.’ … In some of the configurations in which the scanned square is blank (i.e., bears no symbol) the machine writes down a new symbol on the scanned square; in other configurations it erases the scanned symbol. The machine may also change the square which is being scanned, but only by shifting it one place to right or left. In addition to any of these operations the m-configuration may be changed.”

“Turing introduced two fundamental assumptions: discreteness of time and discreteness of state of mind. To a Turing machine, time exists not as a continuum, but as a sequence of changes of state. Turing assumed a finite number of possible states at any given time. “If we admitted an infinity of states of mind, some of them will be ‘arbitrarily close’ and will be confused,” he explained. “The restriction is not one which seriously affects computation, since the use of more complicated states of mind can be avoided by writing more symbols on the tape.”

“The Turing machine thus embodies the relationship between an array of symbols in space and a sequence of events in time. All traces of intelligence were removed. The machine can do nothing more intelligent at any given moment than make a mark, erase a mark, and move the tape one square to the right or to the left. The tape is not infinite, but if more tape is needed, the supply can be counted on never to run out. Each step in the relationship between tape and Turing machine is determined by an instruction table listing all possible internal states, all possible external symbols, and, for every possible combination, what to do (write or erase a symbol, move right or left, change the internal state) in the event that combination comes up. The Turing machine follows instructions and never makes mistakes. Complicated behavior does not require complicated states of mind. By taking copious notes, the Turing machine can function with as few as two internal states. Behavioral complexity is equivalent whether embodied in complex states of mind (m-configurations) or complex symbols (or strings of simple symbols) encoded on the tape.

“It took Turing only eleven pages of ‘On Computable Numbers’ to arrive at what became known as Turing’s Universal Machine. “It is possible to invent a single machine which can be used to compute any computable sequence,” he announced. The Universal Machine, when provided with a suitably encoded description of some other machine, executes this description to produce equivalent results. All Turing machines, and therefore all computable functions, can be encoded by strings of finite length. Since the number of possible machines is countable but the number of possible functions is not, noncomputable functions (and what Turing referred to as “uncomputable numbers”) must exist.

“Turing was able to construct, by a method similar to Gödel’s, functions that could be given a finite description but could not be computed by finite means. One of these was the halting function: given the number of a Turing machine and the number of an input tape, it returns either the value 0 or the value 1 depending on whether the computation will ever come to a halt. Turing called the configurations that halt ‘circular’ and the configurations that keep going indefinitely ‘circle free,’ and demonstrated that the unsolvability of the halting problem implies the unsolvability of a broad class of similar problems, including the Entscheidungsproblem. Contrary to Hilbert’s expectations, no mechanical procedure can be counted on to determine the provability of any given mathematical statement in a finite number of steps. This put a halt to the Hilbert program, while Hitler's purge of German universities put a halt to Göttingen’s position as the mathematical center of the world, leaving a vacuum for Turing’s Cambridge, and von Neumann’s Princeton, to fill.

“After a full year of work, Turing gave Newman a draft of his paper in April of 1936. “Max’s first sight of Alan’s masterpiece must have been a breathtaking experience, and from this day forth Alan became one of Max's principle protégés,” says William Newman, Max's son. Max Newman lobbied for the publication of ‘On Computable Numbers, with an Application to the Entscheidungsproblem,’ in the Proceedings of the London Mathematical Society, and arranged for Turing to go to Princeton to work with Alonzo Church. “This makes it all the more important that he should come into contact as soon as possible with the leading workers on this line, so that he should not develop into a confirmed solitary,” Newman wrote to Church.

“Turing arrived in Princeton carrying his sextant, and stretching his resources to survive on his King’s College fellowship (of £300) for the year. The page proofs of ‘On Computable Numbers’ arrived by mail from London on October 3. “It should not be long now before the paper comes out,” he wrote to his mother on October 6. The publication of ‘On Computable Numbers’ (on November 30, 1936) went largely unnoticed. “I was disappointed by its reception here,’ Turing wrote to his mother in February 1937, adding that “I don’t much care about the idea of spending a long summer in this country.” Only two requests for reprints came in. Engineers avoided Turing’s paper because it appeared entirely theoretical, and theoreticians avoided it because of the references to paper tape and machines …

“In March of I937, Alonzo Church reviewed ‘On Computable Numbers’ in the Journal of Symbolic Logic, and coined the term Turing machine. “Computability by a Turing machine,” wrote Church, “has the advantage of making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately.” Church’s thesis – equating computability with effective calculability – would be the Church-Turing thesis from then on.

“Even Gödel, who dismissed most attempts to strengthen his own results, recognized the Church-Turing thesis as a major advance. “With this concept one has for the first time succeeded in giving an absolute definition … not depending on the formalism chosen,” he admitted in 1946. Before Church and Turing, the definition of mechanical procedure was limited by the language in which the concept was defined. “For the concept of computability however … the situation is different,” Gödel observed. “By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion.”

“It is difficult today to realize how bold an innovation it was to introduce talk about paper tapes and patterns punched in them, into discussions of the foundations of mathematics,” Max Newman recalled in 1955. For Turing, the next challenge was to introduce mathematical logic into the foundations of machines. “Turing’s strong interest in all kinds of practical experiment made him even then interested in the possibility of actually constructing a machine on these lines.”

“The title ‘On Computable Numbers’ (rather than ‘On Computable Functions’) signaled a fundamental shift. Before Turing, things were done to numbers. After Turing, numbers began doing things. By showing that a machine could be encoded as a number, and a number decoded as a machine, ‘On Computable Numbers’ led to numbers (now called ‘software’) that were ‘computable’ in a way that was entirely new” (Dyson, Turing’s Cathedral, pp. 246-250).

Origins of Cyberspace 394; Tomash and Williams T61, T62.

Pp. 230-265 in Proceedings of the London Mathematical Society, series 2, vol. 42, part 3, November 30, 1936 & part 4, December 23, 1936; [with:] ‘... A correction,’ pp. 544-546 in ibid., vol. 43. Three vols., large 8vo, pp. 161-240; 241-320; [iv], 546. The two journal issues in original printed wrappers, preserved in a cloth folding case (remnants of publisher’s printed label on front wrapper of part 3, edges of wrappers of part 4 a little dust-soiled); the journal volume in (probably publisher’s) green cloth, covers ruled in blind, spine lettered in gilt, virtually mint.